Unit  4
Applications of Partial Differentiation
If u and v be continuous and differentiable functions of two other independent variables x and y such as , then we define the determine as Jacobian of u, v with respect to x, y Similarly , JJ’ = 1 Actually Jacobins are functional determines Ex.
ST 4. find 5. If and , find 6. 7. If 8. If , , JJ1 = 1 If , JJ1=1 Jacobian of composite function (chain rule) Then Ex.
Where 2. If and Find 3. If Find
Jacobian of Implicit function
Ex. If If Find
Partial derivative of implicit functions
Ex. If and Find If and Find Find
If Find

Let f(x, y) be a continuous function of x & y. Let be the increments x & y resp. Then the value of f(x, y) will be Expanding above expression by taylor’s theorem & since are very small Hence their product and higher powers are negligible and hence neglected we get, i.e. Similarly, If f be a function of variables x, y, z, t, ………. Then we have Note that
Ex. Q1) Find the percentage error in the area of an ellipse where error of ly is made in measuring it’s major and minor axes. S1)
Q2) The density of a body is calculated from its’s weight W in air and win water if error are made in , find the error in . S2)
Q3) Find the percentage error in computing the parallel resistance r of three resistances r1, r2, r3 from the formula. Where r1, r2, r3 are each in error of 1 & 2y.
Q4) Find the approximate value of S4)

Note that
Ex. Discuss the stationary values of
Ex. Find the values of x and y for which x2 + y2 + 6x = 12 has a minimum values and find its minimum value. Divide 120 into three parts so that the sum of their product. Taken two at a times shall be maximum. Using Lagrange’s method divide 24 into three parts. Such that continued product of the first, square of second, cube of third may be maximum. Find the maximum and minimum value of x2 + y2 when 3x2 + 4xy + 6y2 = 140 is satisfied.

Let be a function of x, y, z which to be discussed for stationary value. 
Let be a relation in x, y, z 
for stationary values we have, 
i.e. … (1) 
Also from we have 
… (2) 
Let ‘’ be undetermined multiplier then multiplying equation (2) by and adding in equation (1) we get, 
… (3) 
… (4) 
… (5) 
Solving equation (3), (4) (5) & we get values of x, y, z and . 

Q1) Decampere a positive number ‘a’ in to three parts, so their product is maximum
S1)
Let x, y, z be the three parts of ‘a’ then we get. 
… (1) 
Here we have to maximize the product 
i.e. 
By Lagrange’s undetermined multiplier, we get, 
… (2) 
… (3) 
… (4) 
i.e. 
… (2)’ 
… (3)’ 
… (4) 
And 
From (1) 
Thus . 
Hence their maximum product is . 

Q2) Find the point on plane nearest to the point (1, 1, 1) using Lagrange’s method of multipliers.
S2)
Let be the point on sphere which is nearest to the point . Then shortest distance. 
Let 
Under the condition … (1) 
By method of Lagrange’s undetermined multipliers we have 
… (2) 
… (3) 
i.e. & 
… (4) 
From (2) we get 
From (3) we get 
From (4) we get 
Equation (1) becomes 
i.e. 
y = 2 
If where x + y + z = 1. 
Prove that the stationary value of u is given by, 

Reference Books:
1. Advanced Engineering Mathematics by Erwin Kreyszig (Wiley Eastern Ltd.)
2. Advanced Engineering Mathematics by M. D. Greenberg (Pearson Education)
3. Advanced Engineering Mathematics by Peter V. O’Neil (Thomson Learning)
4. Thomas’ Calculus by George B. Thomas, (AddisonWesley, Pearson)
5. Applied Mathematics (Vol. I & Vol. II) by P.N.Wartikar and J.N.Wartikar Vidyarthi Griha Prakashan, Pune.
6. Linear Algebra –An Introduction, Ron Larson, David C. Falvo (Cenage Learning, Indian edition)